"We are getting phone calls, and they will probably become a little bit more voluminous if we don't turn it around pretty quickly," White Sox general manager Rick Hahn said.

Heyman's sources indicate that Alex Rios and Alexei Ramirez will at least be discussed, with one rival general manager telling him that Rios will draw more interest. Rios,32, is owed $12.5MM in 2014, the final guaranteed year of his contract, though his deal contains a club option as well. Ramirez, 31, is controlled through 2015 and owed $9.5MM in 2014, $10MM in 2015 and has a $1MM buyout on his $10MM club option for 2016.

Heyman also speculates that names such as Jake Peavy, Jesse Crain, Matt Thornton, Matt Lindstrom and perhaps even Gordon Beckham will become available if the White Sox do indeed take the position of sellers this July. Crain, of course, is on a historic run, having fired 28 shutout innings over his past 28 appearances with a 37-to-7 K/BB ratio in that time.